Write the sum $LaTeX: \displaystyle -4-3-2 \ldots +20+21$ in sigma notation and then find the sum.

The common difference is given by $LaTeX: \displaystyle a_2-a_1=-3-(-4)=1$ . Using the first term gives the sequene $LaTeX: \displaystyle a_n= -4+(n-1)(1)$ . Setting the general term equal to the last term and solving for $LaTeX: \displaystyle n$ gives $LaTeX: \displaystyle -4+(n-1)(1)=21 \implies n = 26$ . Writing in sigma notation gives $LaTeX: \displaystyle \displaystyle \sum_{n=1}^{26} \left(n - 5\right)$ . Using the formula for a finite arithmetic sum gives $LaTeX: \displaystyle \frac{ 26(-4+21) }{2}=221$ .